Steady states of FitzHugh-Nagumo-type systems with sign-changing coefficients

We establish existence and multiplicity results for steady-state solutions of spatially heterogeneous FitzHugh-Nagumo-type systems, extending the existing theory from constant to variable coefficients that may change sign. Specifically, we study the system Specifically, we study the system \begin{align*} -Δu + a(x)v &= f(x,u) && $\text{in }$ $\mathbb{R}$^N, \ -Δv + b(x)v &= c(x)u && $\text{in }$ $\mathbb{R}$^N. \end{align*} where $N⩾3$, the coefficients $a,b,c:{\mathbb{R}}^N\to \mathbb{R}$ are $L_{\mathrm{l}\mathrm{o}\mathrm{c}}^{\mathrm{\infty }}$-functions bounded from below, and $f:{\mathbb{R}}^N\times \mathbb{R}\to \mathbb{R}$ is a Carathéodory function with subcritical growth. For assumptions permitting sign changes and non-coercivity of the coefficients, we prove the existence of a mountain pass solution. In the case where $a,b,c$ do not change sign, still allowing non-coercive behavior, we additionally establish the existence of componentwise positive and negative solutions.